Gábor Braun

Approximation Limits of Linear Programs (Beyond Hierarchies)

We develop a framework for proving approximation limits of polynomial size linear programs (LPs) from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any LP as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n1/2-ϵ)-approximations for CLIQUE require LPs of size 2nΩ(ϵ). This lower bound applies to LPs using a certain encoding of CLIQUE as a linear optimization problem. Moreover, we establish a similar result for approximations of semidefinite programs by LPs. Our main technical ingredient is a quantitative improvement of Razborov’s [38] rectangle corruption lemma for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix.

The matching polytope does not admit fully-polynomial size relaxation schemes

The groundbreaking work of Rothvo{\ss} [arxiv:1311.2369] established that every linear program expressing the matching polytope has an exponential number of inequalities (formally, the matching polytope has exponential extension complexity). We generalize this result by deriving strong bounds on the polyhedral inapproximability of the matching polytope: for fixed

Surgery formula for Seiberg–Witten invariants of negative definite plumbed 3-manifolds

0 < \varepsilon < 1

Invariants of Newton non-degenerate surface singularities

, every polyhedral

Information-theoretic approximations of the nonnegative rank

(1 + \varepsilon / n)

The matching problem has no small symmetric SDP

-approximation requires an exponential number of inequalities, where

A Polyhedral Characterization of Border Bases

n

The Matching Problem Has No Fully Polynomial Size Linear Programming Relaxation Schemes

is the number of vertices. This is sharp given the well-known

Strong Reductions for Extended Formulations

\rho

Info-greedy sequential adaptive compressed sensing

-approximation of size